openings design in underground mines, different approaches, kirscha formulae for circular opening, plastic xzone effect on stability of opening, radial and tangential stresses distribution

1. DESIGN OF OPENINGS
U.Siva Sankar
Sr. Under Manager
Project Planning
Singareni Collieries Company Ltd
E-Mail :ulimella@gmail.com or
uss_7@yahoo.com
Visit at:
www.slideshare.net/sankarsulimella
THE MINING ENVIRONMENT
IN-SITU STRESSES
1

2. Stress beneath surface
Y
σy
τxy
σx
r y
θ
X
x
Natural stresses which exist prior to excavation. The natural stress field can be
composed of gravitational stress, tectonic stresses and residual stresses.
Induced stresses are man-made stress component due to removal or addition of
material.
2

3. Ground Control ???
No excavation No Problem
All rock in the ground
subjected to compressive
stresses and if excavation is
made the rock left standing
has to take more load
because the original
support provided by the
rock within the excavation
has been removed.
Effect of vertical force line on u/g
opening
• An underground opening brings about a lateral squeeze
of vertical force lines. This results in a large stress
concentration at the abutments a and b.
3

4. Induced stresses
• It is man-made stress component due to removal or
addition of material.
• The reasons for induced stress conditions in rock are:
– Loosening of the rock mass around cavities;
– Weight of the overburden rock mass;
– Tectonic forces, and
– Volumetric expansion of the rock mass by thermal effects, or by
swelling brought about by the action of physical or physico-
chemical processes.
• Depending upon the kind and properties of the rock,
induced stress conditions require the use of different
construction methods. All of the above-mentioned four
induced stress (pressure) conditions may also occur
simultaneously.
Division of area around an
excavation
• Near Field
– Adjacent to the excavations
– Area of interest to the designer
• Far Field
– Remote from excavation
– The response of the rock is essentially elastic
4

5. Requirements for Stress
Analysis??
– Rock Mass Property
– Geological Discontinuities
• Fracture patterns, Fault zones, Joints, Bedding
planes etc
– Excavation Geometry
Stress Assumptions
• Prior to disturbance of rock mass, it is visualized
that the rock is subjected to the effect of the
rock-forming forces as well as the force of
gravity.
• Vertical stresses are equal to σv = ρgh
• The horizontal stresses are generated by
assuming that the strain is zero.
σh = (ν/1-ν)σv =k σv
For example: ν = 0.3; σh = 0.43 σv
5

6. Vertical and Horizontal stresses
Vertical Stress (after Brown Townend and Zoback, (2000)
and Hoek, 1978)
Ratio of Horizontal to Vertical Stress
Sheory,1994
 1
K = 0.25 + 7 Ek  0.001 + 
 z
where Ek (GPa) is the average deformation modulus of the upper part of the
earth’s crust measured in a horizontal direction.
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7. Stresses around mine openings
• The underground rock structures (openings) are
the excavations which are created in a pre-
stressed environment. Stress analysis provide
insight into the changes in preexisting stress
equilibrium caused by an opening. It interprets
the performance of an opening in terms of stress
concentrations and associated deformations and
serves as a rational basis for establishing the
performance of requirements for design.
• The properties of the rockmass are complex,
and no single theory is available to explain
rockmass behaviour. However, the theories of
elasticity and plasticity provide results that have
relevance to the stress distributions induced
about openings and provide a first step to
estimating the distribution of stresses around
openings.
• Prior to excavation, the in situ stresses in the
rockmass are in equilibrium. Once the
excavation is made, the stresses in the vicinity of
the opening are redistributed and stress
concentrations develop.
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8. Design Approach
• Analytical solution
• Empirical Solution
• Physical Modeling
• Numerical Modeling
Various cross section of
Underground openings
Circular Analytical solution by Kirsch in 1898.
Elliptical Analytical solution by Bray in 1976.
Parabolic cross-section Square
Horse- Shoe Shape
Rectangular
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9. Circular Opening
• Prediction of the stresses and displacements
around a circular opening in the rock mass at
great depth is an important problem in
geotechnical, petroleum and mining engineering
such as the design of tunnels, boreholes and
mine shafts.
Stress concentration around mine openings
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10. Kirsch Equation (1898)
• Analytical or Closed form solution for
circular opening in 2D.
• Continuous, Homogenous, Isotropic and
Linear Elastic material (CHILE).
• Circular opening of radius, a
• In Polar co-ordinate system find the radial,
tangential and shear stress as well as
radial and tangential displacement at a
point located at (r,θ).
Krisch’s equations for stresses around mine
openings
1 
  a2   a2 a4  

σ r = σ z (1 + k ) 1 − 2 
− (1 − k ) 1 − 4 2 + 3 4  cos 2θ 
2 
  r   r r  

1    a2   a4  

σ θ = σ z (1 + k ) 1 + 2  + (1 − k ) 1 + 3 4  cos 2θ 
2    r   r  

1    a2 a4  

τ rθ = σ z (1 − k ) 1 + 2 2 − 3 4  sin 2θ 
2    r r  

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11. Inferences from kirsch equation
For any excavation shape, only tangential stresses exist at the boundary of
an excavation (without internal loading), for example: put a/r = 1
For, r = 4a, the ratio of induced to applied stress are close to unity,
means no influence of opening beyond that.
For a hydrostatic stress field (k =1),
– The stress concentration is 2 times SV or SH everywhere on the
boundary.
– No, shear stress any where within rock mass.
For k = 0, maximum stress concentration is 3 (i.e. compression) and
minimum stress concentration is -1(Tensile).
Stress is independent of elastic constant like young’s modulus and poisons
ratio.
The equation is not including the radius, but a ratio a/r (dimensionless)
(dimensionless)
i.e. the stress at boundary of an excavation are independent of
absolute value of the radius.
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12. Fig: Stress concentration along the wall and roof of a rectangular
opening
The stress concentrations along the perimeter of an
opening show a large increase for non-circular openings or
angular openings.
Hoek & Brown -determined tangential stresses on the excavation surface at the crown
.
and in the sidewall for different –shaped openings for a range of in situ stress ratios
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13. Importance of Elastic analysis
• Maximum and minimum stresses on the
boundary of the opening
• Boundary displacement induces by the
excavation
• Extent of zone of influence
• The extent of the overstresses region
• The increase in strain energy, and the
dynamic energy released, when an
excavation is generated
Extent of Plastic Zone
13

14. Zone of Disturbance
Upon blasting out an opening,
the rock becomes
disintegrated.
Loose zone, or zone of
disturbance between the
contour of the opening and
the plastic zone, is created.
Because of the disintegrated
condition of the rock, the
rock here is not in a plastic
state. It does not have any
unconfined compressive
strength, nor can it take and
carry any stress.
Elastic and plastic zones around circular opening
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15. Design methods for single
opening
• Any opening will be stable if the maximum stress occurring around
the opening is less than the failure strength of the rocks defined in a
failure criterion.
• Simplest method of designing is to determine what type of opening
in what geometry produces the maximum stress and compare it with
the failure strength of the rocks.
Stress Shadow
• First case shows
stresses amplified
between excavations.
• Second case shows
stresses attenuation.
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16. Zone of Influence
• The zone around an excavation in which the
stresses are perturbed from their in-situ
values by more than a defined amount.
• For example, we could define the zone of
influence around the excavation as the zone
within which at least one component of the
stress tensor is perturbed by greater than, say,
5% of its in situ value, expressed mathematically
as
σ Induced − σ Natural ≥ 0.05σ Natural
r5% ≤ a 20
Excavation sequencing alternatives
Should we
create
excavation I
or excavation
II first ?
• Advantage of creating smaller dia opening II first is that the final stress field
acting on Excavation I will be in place before that excavation is made, and the
process of creating Excavation I will not appreciably affect Excavation II.
• Advantage of creating large dia Excavation I first is that the excavation is made
in an undisturbed stress field, and the tunnel can be supported in anticipation of
the stresses that will be induced following the creation of Excavation II.
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17. Conclusion
• For the analysis of stability of underground
openings, the knowledge of stresses,
strength and failure mechanism are
important. The idea of the stress
concentrations and their effects on the
surroundings of the openings helps the
design engineers to plan a suitable
method of support system. However the
knowledge of rockmass properties are still
to be acquired and rockmass classification
systems is an attempt towards the
purpose.
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